Surface automorphisms and elementary number theory

The modular surface $\mathbb{H}/\Gamma,\, \Gamma= \mathrm{SL}(2,\mathbb{Z})$ has many covers - for example the three punctured torus $\mathbb{H}/\Gamma(2)$ and the once punctured torus $\mathbb{H}/\Gamma'$. We will discuss how classical Diophantine approximation can be interpreted in terms of the behaviour of geodesics on the once punctured torus and a geometric reformulation of the Frobenius uniqueness conjecture.
We will then give an account of two theorems of Fermat in terms of   the automorphisms of $\mathbb{H}/\Gamma(2)$:
- if $p$ is a prime such that $4|(p-1)$ then  can be written as a   sum of squares $p = c^2 + d^2$

- if $p$ is a prime such that $3|(p-1)$ then  can be written as  $  p = c^2 +cd +  d^2$
Finally we will discuss possible extensions to surfaces of the for  m $\mathbb{H}/\Gamma_0(N)$.